Receivers based on closed-form parametric estimates of the probability density function for the received signal

ABSTRACT

A closed-form parametric approach to channel-estimation is provided. In one aspect, a specific parametric expression is presented for the received signal pdf that accurately models the behavior of the received signal in IM/DD optical channels. The corresponding parametric channel-estimation approach simplifies the design of MLSE receivers. The general technique lends itself well to the estimation of the signal pdf in situations where there are multiple sources of noise with different distributions, such as ASE noise, together with Gaussian and quantization noise, and signal-dependent noise, for example.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This application claims priority under 35 U.S.C. §119(e) to U.S.Provisional Patent Application Ser. No. 60/821,137, “Parametricestimation of IM/DD optical channels using new closed-formapproximations of the signal pdf,” filed Aug. 2, 2006. The subjectmatter of the foregoing is incorporated herein by reference in itsentirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates generally to receivers and, more specifically, toreceivers that use closed-form parametric estimates of the channeland/or received signal probability density function.

2. Description of the Related Art

In many situations, receiver performance depends on knowledge about thereceived signal and/or the channel over which the signal has propagated.For example, in the case of intensity-modulation/direct-detection(IM/DD) optical-transmission systems at speeds of 10 Gb/s and higher,chromatic dispersion and polarization-mode dispersion have become majorfactors that limit the reach of these systems. Electronic dispersioncompensation (EDC) is an increasingly popular approach to mitigate theseimpairments and a cost-effective alternative to purelyoptical-dispersion-compensation techniques.

Among EDC techniques, maximum-likelihood sequence estimation (MLSE) is apromising approach. MLSE chooses the sequence that minimizes thenegative logarithm of the likelihood function (i.e., the metric).However, MLSE receivers require knowledge of the statistics of the noisyreceived signal. Noise in IM/DD optical channels is stronglynon-Gaussian and signal dependent. Except in the simplest situations,the pdf of the signal corrupted by noise does not have a closed-formexpression. This can lead to difficulties in the implementation of theMLSE receiver.

If the signal pdf is not known a priori by the receiver, it must beestimated based on the received signal. This is a process known aschannel-estimation. In an EDC receiver implemented as a monolithicintegrated circuit, channel-estimation algorithms typically must beimplemented by dedicated hardware. The amount of computational resourcesthat can be devoted to channel-estimation is usually limited byconstraints on the chip area and power dissipation. Therefore, findingcomputationally efficient channel-estimation methods is of paramountimportance.

Channel-estimation methods can be parametric or nonparametric.Parametric methods assume that the functional form for the pdf of thesignal is known but its parameters are not, whereas nonparametricmethods do not assume any knowledge of the pdf. The main difficulty withnonparametric methods is that a large number of samples are needed toobtain accurate estimates. This is particularly problematic in the tailregions of the signal pdf, where it may take an inordinate amount oftime to obtain enough samples. For this reason, parametric methods arepreferable. However, parameter estimation may be difficult if thefunctional form assumed for the pdf is cumbersome or does not have aclosed-form expression, particularly when the estimation must be done byhardware operating in real time, as in the case of an adaptive EDCreceiver.

Thus, there is a need for improved, computationally efficient approachesto channel estimation and receivers that depend on channel-estimation.

SUMMARY OF THE INVENTION

The present invention overcomes the limitations of the prior art byproviding a closed-form parametric approach to channel-estimation. Inone aspect, a specific parametric expression is presented for thereceived signal pdf that accurately models the behavior of the receivedsignal in IM/DD optical channels. The corresponding parametricchannel-estimation approach simplifies the design of MLSE receivers. Thegeneral technique lends itself well to the estimation of the signal pdfin situations where there are multiple sources of noise with differentdistributions, such as ASE noise, together with Gaussian andquantization noise, and signal-dependent noise, for example.

Another aspect of the invention is the computation of bit-error rates(BER), for example for MLSE receivers operating on IM/DD channels. As aspecific example, a closed-form analytical expression for the bit-errorprobability of MLSE-based receivers in dispersive optical channels inthe presence of ASE noise and post detection Gaussian noise ispresented. Analytical expressions of the BER are useful not only topredict system performance, but also to facilitate the design of channelcodes. Numerical simulations demonstrate the accuracy of the closed-formparametric expressions.

One aspect of the invention includes a receiver based on closed-formparametric channel estimation. In one specific embodiment, the receiverincludes a parametric channel-estimator, a branch-metric computationunit and a decoder (e.g., a Viterbi decoder). The parametricchannel-estimator provides a channel-estimate based on a closed-formparametric model of the channel. The parameters for the model areestimated based on the received signal. The branch-metric computationunit determines branch metrics for each of the possible received bitsequences based in part on the channel-estimate from the parametricchannel-estimator, and the decoder determines the received bit sequencebased in part on the branch metrics from the branch-metric computationunit. One advantage of closed form parametric estimates is that theytypically require fewer computational resources to implement. Theparametric approach can also be used in other embodiments for otherchannels (e.g., noise sources) and/or receiver architectures (e.g.,different types of decoders).

Other aspects of the invention include methods and systems correspondingto the devices described above.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention has other advantages and features which will be morereadily apparent from the following detailed description of theinvention and the appended claims, when taken in conjunction with theaccompanying drawings, in which:

FIG. 1 is a block diagram of an example system suitable for use with thepresent invention.

FIGS. 2A-2C are graphs of parameter v_(s), as a function of OSNR fordifferent values of r₁₀.

FIG. 3 is a graph of AME as a function of OSNR, for various channelestimation methods, values of SGNR and values of r₁₀.

FIGS. 4A-4B are graphs of the signal pdf, for various worst casescenarios.

FIGS. 5A-5B are graphs of AME as a function of OSNR, for various channelestimation methods, values of SGNR and values of r₁₀.

FIG. 6A is a graph of AME as a function of the extinction-ratio r₁₀ forvarious channel estimation methods.

FIG. 6B is a graph of BER as a function of r₁₀ for various channelestimation methods.

FIGS. 7A-7B are graphs of AME as a function of OSNR, for various channelestimation methods, values of SGNR and values of L.

FIGS. 8A-8B are graphs of signal pdf's for the parametric channelestimate and Gaussian approximation, respectively.

FIGS. 9A-9B are graphs of signal pdf's for various channel estimationmethods, using realistic filters.

FIG. 10 is a further graph of signal pdf's according to the invention.

FIGS. 11A-11C are graphs of log(BER) as a function of OSNR, for variousvalues of SGNR and r₁₀.

FIGS. 12A-12B are graphs of log(BER) as a function of OSNR, for variousvalues of SGNR and D.

FIGS. 13A-13B are graphs of log(BER) as a function of OSNR, for variousvalues of SGNR and M.

FIG. 14 is a block diagram of an MLSE receiver according to the presentinvention.

FIG. 15 is a flow diagram for updating the LUTs shown in FIG. 14.

The figures depict embodiments of the present invention for purposes ofillustration only. One skilled in the art will readily recognize fromthe following discussion that alternative embodiments of the structuresand methods illustrated herein may be employed without departing fromthe principles of the invention described herein.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Outline

-   1. MLSE in IM/DD Optical Channels    -   1.A. A Closed-Form Approximation for the Signal PDF    -   1.B. MLSE Receiver-   2. Example Closed-Form Parametric Expression for the Signal PDF    -   2.A. Approximating T_(s)(•) in IM/DD Optical Channels    -   2.B. Channel-estimation    -   2.C. On the Gaussian Approximation (GA)-   3. Accuracy of the Parametric Expression for Signal PDF (22)    -   3.A. Measure of the Goodness of PDF Approximations for Metrics        Evaluation    -   3.B. Channel Estimation in the Presence of Ideal Filters    -   3.C. Channel Estimation in the Presence of Ideal Filters and        Quantization    -   3.D. Channel Estimation in the Presence of Realistic Filters-   4. Performance of MLSE in IM/DD Optical Channels    -   4.A. Closed-Form Approximation for the Error-Event Probability        in Dispersive IM/DD Channels    -   4.B. Numerical Results-   5. Example Implementations    -   5.A. Practical Implementation using the Method of Moments-   6. Further Examples    1. MLSE in IM/DD Optical Channels

Various aspects of the invention will first be introduced in the contextof MLSE receivers for IM/DD systems. FIG. 1 shows a simplified model ofsuch a system. The transmitter 110 modulates the intensity of thetransmitted signal using a binary alphabet (e.g., ON-OFF-keying (OOK)modulation). Let {a_(n)}=(a₁, a₂, . . . , a_(N)) and N denote,respectively, the bit sequence to be sent on the optical fiber 120(a_(n)ε{0,1}) and the total number of transmitted symbols. The opticalpower ratio between the pulses representing a logical 1 and a logical 0,r₁₀, is called the extinction ratio. Assume that the intensity level fora logical 0 (a_(n)=0) is different from zero, which is usual inpractical transmitters (e.g., r₀₁=r₁₀ ⁻¹≈0.1). The optical fiberintroduces chromatic dispersion and polarization mode dispersion, aswell as attenuation. Optical amplifiers 122 are deployed periodicallyalong the fiber 120 to compensate the attenuation, also introducing ASE(amplified spontaneous emission) noise in the signal. ASE noise ismodeled as additive white Gaussian noise in the optical domain. At thereceiver 130 (elements 132-140), the optical signal is filtered byoptical filter 132 and then converted to a current with a p-i-n diode134 or avalanche photodetector 134. The resulting photocurrent isfiltered by an electrical filter 136. For simplicity, in the followinganalysis, we use an ideal low-pass optical filter 132 and anintegrate-and dump electrical filter 136. The output of the filter 136is ideally sampled 138 (i.e., with infinite resolution) at the symbolrate 1/T and applied to the MLSE 140.

The received samples can be written asy _(n) =s _(n) +r _(n) +z _(n) =x _(n) +z _(n)  (1)where s_(n)=ƒ(a_(n), . . . , a_(n−δ+1)) is the noise-free signal in theelectrical domain, which is, in general, a nonlinear function of a groupof δ consecutive transmitted bits (note that s_(n)εS={ s ₀, s ₁, . . . ,s _(k)} with s ₀=ƒ(0, 0 . . . , 0), s ₁=ƒ(0, 0 . . . , 1), . . . , s_(k)=ƒ(1, 1 . . . , 1)) where K=2^(δ−1); r_(n) are samples of the ASEnoise in the electrical domain; x_(n)=s_(n)+r_(n); and z_(n) are samplesof the electrical (thermal) noise, which is modeled as a zero-meanGaussian random process.

Then, the pdf's of x and z can be expressed as

$\begin{matrix}{{f_{x|s}\left( x \middle| s \right)} = {\frac{1}{N_{0}}\left( \frac{x}{s} \right)^{\frac{M - 1}{2}}{\mathbb{e}}^{- \frac{x + s}{N_{0}}}{I_{M - 1}\left( \frac{2\sqrt{x\; s}}{N_{0}} \right)}}} & (2) \\{{f_{z}(z)} = {\frac{1}{\sqrt{2{\pi\sigma}_{e}}}{\mathbb{e}}^{- \frac{z^{2}}{2\sigma_{e}^{2}}}}} & (3)\end{matrix}$where σ_(e) ² is the power of z, N₀ is related to the variance of theASE noise in the optical domain, M is the ratio of the optical toelectrical bandwidth of the receiver, and I_(m)(•) is the m^(th)modified Bessel function of the first kind For convenience, the timeindex n is omitted. Note that the ASE noise component in thepolarization orthogonal to the signal is neglected in (2) and that thechi-square pdf for the ASE noise is not exact in the presence ofpractical optical/electrical filters. However, the analysis andassumptions presented in the following are still valid, even whenpractical filters are used.

Since the noise components r and z are independent random variables, theconditional pdf of y can be obtained from (2) and (3) as follows:ƒ_(y|s)(y|s)=ƒ_(x|s)(y|s)

ƒ_(z)(y), sεS  (4)where

denotes convolution.

1.A. A Closed-Form Approximation for the Signal PDF

The signal pdf (4) does not have a closed-form analytical expression.However, when r₀₁>0 and N₀ is sufficiently small, it can be approximatedusing the following expression for (2):

$\begin{matrix}{{f_{x|s}\left( x \middle| s \right)} \approx {\frac{1}{2\sqrt{\pi\overset{\sim}{s}N_{0}}}{\mathbb{e}}^{- \frac{{({\sqrt{x} - \sqrt{\overset{\sim}{s}}})}^{2}}{N_{0}}}}} & (5)\end{matrix}$with x≧0 and{tilde over (s)}=E _(s) {x}=s+I _(sp)  (6)where E_(s){•} denotes conditional expectation given the noise-freesignal level s, and I_(sp)=E_(s){r_(n)}=N₀M. When N₀<<s_(n), notice that{tilde over (s)}≈s. Then, using (5) in pdf (4) and applying the methodof steepest descent to approximate the convolution integral, the pdf (4)can be expressed as:

$\begin{matrix}{{f_{y|s}\left( y \middle| s \right)} \approx {G_{s}{\mathbb{e}}^{- {g_{s}{(y)}}}}} & (7) \\{{g_{s}(y)} = {\frac{\left( {\sqrt{w} - \sqrt{s}} \right)^{2}}{N_{0}} + \frac{\left( {y - w} \right)^{2}}{2\sigma_{e}^{2}}}} & (8)\end{matrix}$where G_(s), is a signal-dependent factor such ∫_(−∞) ^(∞)G_(s)e^(−g)^(s) ^((y))dy=1, while ω>0 is the value that minimizes (8).

1.B. MLSE Receiver

The maximum-likelihood sequence receiver for signals affected bynonlinear intersymbol interference and additive Gaussian noise consistsof a matched-filter bank followed by a Viterbi decoder. It is known thatin the case of Gaussian noise, samples of the signal taken at the outputof the matched filter at the symbol rate constitute a set of sufficientstatistics for the detection. In the case of non-Gaussian andsignal-dependent noise, the problem of obtaining a set of sufficientstatistics by sampling a filtered version of the input signal at thesymbol rate has not been solved. In the following, we assume that theoutput of the photodetector is filtered and then sampled at the symbolrate, but we do not assume that the input filter is a matched filterbank. We assume that the samples of the signal plus noise areindependent, but they are not identically distributed.

The MLSE receiver chooses, among the 2^(N) possible sequences, the one{â_(n)}=(â₁, . . . , â_(N)) that minimizes the cumulative metric

$\begin{matrix}{M = {\sum\limits_{n = 1}^{N}\;{- {\ln\left( {f_{y|s}\left( y_{n} \middle| {\hat{s}}_{n} \right)} \right)}}}} & (9)\end{matrix}$where ŝ_(n)=ƒ(â_(n), . . . , â_(n−δ+1)). The minimization can beefficiently implemented using the Viterbi algorithm. Whereas in Gaussianchannels, the branch metrics are simple Euclidean distances; in theseoptical channels, the branch metrics require the evaluation of differentfunctions for each branch. This is the result of the fact that the noiseis signal dependent. In general, the functions representing the branchmetrics do not have a closed-form analytical expression.2. Example Closed-Form Parametric Expression for the Signal PDF

It is well known from the literature that the random variable y can betransformed into a Gaussian random variable u by using a nonlineartransformation T_(s)(•) as follows:u=T _(s)(y)=F _(u|s) ⁽⁻¹⁾(F _(y|s)(y))  (10)where F_(u|s)(•) and F_(y|s)(•) are the cumulative distributionfunctions of u and y, respectively, when the noise-free signal s isreceived. From (10) it is possible to show that

$\begin{matrix}{{{f_{y|s}\left( y \middle| s \right)} = {\frac{1}{\sqrt{2{\pi\varsigma}_{s}}}{\mathbb{e}}^{- {\frac{1}{2\varsigma_{s}}{\lbrack{{T_{s}{(y)}} - {\overset{\_}{u}}_{s}}\rbrack}}^{2}}{T_{s}^{\prime}(y)}}},{\forall{y.}}} & (11)\end{matrix}$where T′_(s)(y)=dT_(s)(y)/dy, while ū_(s) and ζ_(s) and are the mean andvariance of u, respectively.

Suppose that y is concentrated near its means {tilde over (s)} soƒ_(y|s)(y|s) is negligible outside an interval ({tilde over (s)}−ε,{tilde over (s)}+ε) with ε>0, and in this interval,T′_(s)(y)≈T′_(s)({tilde over (s)}). From the Chebyshev inequality, notethat this condition can be verified when the noise power is sufficientlylow, that is, Pr{|y−{tilde over (s)}|≧ε}≦(M_(2,s)/ε²) with M_(2,s) beingthe conditional second-order central moment of the received signal y.Thus, we can verify thatE _(s) {y}={tilde over (s)}≈s  (12)ū _(s) ≈T _(s)({tilde over (s)})≈T _(s)(s)  (13)ζ_(s) ≈[T′ _(s)({tilde over (s)})]² M _(2,s) ≈[T′ _(s)(s)]² M_(2,s)  (14)T′ _(s)(y)≈T′ _(s)(s)  (15)Using (13)-(15), it is simple to show that the generic pdf (11) can beapproximated by

$\begin{matrix}{{{f_{y|s}\left( y \middle| s \right)} \approx {\frac{1}{\sqrt{2\pi\; M_{2,s}}}{\mathbb{e}}^{- {\frac{1}{2\varsigma_{s}}{\lbrack{{T_{s}{(y)}} - {T_{s}{(s)}}}\rbrack}}^{2}}s}} \in {S.}} & (16)\end{matrix}$

Based on (16), we can verify that minimizing the cumulative metric Mfrom (9) is equivalent to minimizing

$\begin{matrix}{\hat{M} = {{\sum\limits_{n - 1}^{N}\;{\frac{1}{2\varsigma_{s_{n}}}\left\lbrack {{T_{{\hat{s}}_{n}}\left( y_{n} \right)} - {T_{{\hat{s}}_{n}}\left( {\hat{s}}_{n} \right)}} \right\rbrack}^{2}} + {\frac{1}{2}{{\ln\left( {2\pi\; M_{2,{\hat{s}}_{n}}} \right)}.}}}} & (17)\end{matrix}$From (13) and (14) note that, once T_(s)(•) is known, all parametersrequired to evaluate (17) can be directly estimated from the inputsamples.

2.A. Approximating T_(s)(•) in IM/DD Optical Channels

The exact conditional pdf of the received signal can be written asƒ_(y|s)(y|s)=G _(s) e ^(−g) ^(s) ^((y))  (18)where G_(s) is a normalization factor, and g_(s)(•) is a given function.In IM/DD optical channels with combined ASE noise and Gaussian noise,g_(s)(•) can be accurately approximated by (8). From (16) and (18), notethat

$\begin{matrix}{{g_{s}(y)} \approx {{\frac{1}{2\varsigma_{s}}\left\lbrack {{T_{s}(y)} - {T_{s}(s)}} \right\rbrack}^{2}.}} & (19)\end{matrix}$

In general, obtaining a simple analytical expression for T_(s)(•) from(10) or (19) is difficult. However in most cases of interest, it ispossible to derive a good approximation by analyzing the properties ofg_(s)(•) and T_(s)(•). Assuming that T_(s)(•) is a differentiableincreasing function, it can be approximated byT _(s)(y)≈H _(Θs)(y), sεS  (20)where H_(Θs)(y) is a given differentiable increasing function withunknown parameters defined by the set Θ_(s). Additionally, it can beshown that in optical channels with combined ASE noise andpost-detection Gaussian noise, T_(s)(•) is a concave function. Thus,from (20), we conclude that function H_(Θs)(•) should also be concave.Assuming that r₀₁>0 and the signal-to-noise ratio (SNR) is sufficientlyhigh (i.e., Pr{y_(n)<0}≈0)), we have found that the set of parametricconcave functions defined byH _(Θs)(y)=y ^(v) ^(s) , 0<v _(s)≦1  (21)with Θ_(s)={v_(s)} is adequate to accurately approximate T_(s)(y) intransmissions over optical channels. Note that a linear function(v_(s)=1) is both concave and convex. Using (12)-(16) and (21), weobtain the following closed form parametric approximation for the pdf ofthe received signal:

$\begin{matrix}{{{f_{y|s}\left( y \middle| s \right)} = {\frac{v_{s}s^{({v_{s} - 1})}}{\sqrt{2\pi\;\varsigma_{s}}}{\mathbb{e}}^{{- \frac{1}{2\varsigma_{s}}}{({y^{v_{s}} - s^{v_{s}}})}^{2}}}},{s \in {S.}}} & (22)\end{matrix}$where ζ_(s) and v_(s) are defined by (11) and (21).

For combined ASE noise and post-detection Gaussian noise (σ_(e)>0,N₀>0), note that Pr{y≦0}>0; thus approximation (21) may not be definedif v_(s)<1. As we shall show later, this problem can be overcome byadding an appropriate constant y_(c) to the input signal and neglectingthe negative values of y+y_(c).

2.B. Channel-estimation

Parameters N₀, σ_(e) ², M, and set S can be obtained by using the methodof moments (e.g., see O. E. Agazzi, et al., “Maximum Likelihood SequenceEstimation in Dispersive Optical Channels,” J. Lightwave Technology,vol. 23 no. 2, pp. 749-763, February 2005, which is incorporated byreference herein). The sets of parameters {v_(s)} and {ζ_(s)} with sεScan be calculated as follows. Since u=y^(v) ^(s) is approximately aGaussian random variable with mean ū_(s), its third-order central momentη_(s) should be zero for the optimal value of v_(s), that isη_(s) =E{(y ^(v) ^(s) −ū _(s))³}=0.  (23)A simple iterative algorithm can be used to look for the value of v_(s)that minimizes |η_(s)|. The third-order central moment η_(s) can benumerically evaluated by using the signal pdf (4).

In the case of IM/DD optical channels with combined ASE noise andGaussian noise, v_(s) can be estimated directly from the central momentsof y as follows:

$\begin{matrix}{{v_{s} \approx {1 - {\frac{2}{3}\frac{{sM}_{3,s}}{M_{4,s} - M_{2,s}^{2}}}}},{s \in S}} & (24)\end{matrix}$where M_(3,s) and M_(4,s) are the conditional third and fourth-ordercentral moments of the received signal y, respectively. From (24), bothGaussian noise (σ_(e)>0, N₀=0) and ASE noise (σ_(e)=0, N₀>0) are specialcases of (22):

$\begin{matrix}{{{{{Gaussian}\mspace{14mu}{noise}\text{:}\mspace{14mu}\varsigma_{s}} = \sigma_{e}^{2}},{v_{s} = 1},{y \in \Re}}{{{{ASE}\mspace{14mu}{noise}\text{:}\mspace{14mu}\varsigma_{s}} = \frac{N_{0}}{2}},{v_{s} = 0.5},{y > 0.}}} & (25)\end{matrix}$

Once v_(s) is estimated, parameters ζ_(s) and ū_(s) can be obtained from(13) and (14) as follows:ū _(s) ≈s ^(v) ^(s)   (26)ζ_(s) ≈v _(s) ² s ^(2(v) ^(s) ⁻¹⁾ M _(2,s).  (27)

2.C. On the Gaussian Approximation

When the SNR is sufficiently high, the term between brackets in theexponent of (16) can be approximated byT _(s)(y)−T _(s)(s)≈(y−s)T′ _(s)(s).  (28)Then, from (14) and (28), it can be shown that the generic pdf (16)results in

$\begin{matrix}{{{f_{y|s}\left( y \middle| s \right)} \approx {\frac{1}{\sqrt{2\pi\; M_{2,s}}}{\mathbb{e}}^{{- \frac{1}{2M_{2,s}}}{({y - s})}^{2}}}},{s \in S}} & (29)\end{matrix}$and therefore, the generic pdf (16) reduces to the Gaussianapproximation.3. Accuracy of the Parametric Expression for Signal PDF (22)

Approximation (16) for the generic pdf is valid when the noise power issufficiently low, such that conditions (13)-(15) are satisfied. On theother hand, the validity of T_(s)(y)≈y^(v) ^(s) for IM/DD opticalchannels depends directly on the accuracy of (5), which has been foundto be satisfactory when r₀₁>0 and OSNR{tilde under (>)}5 dB. Our resultsshow that the parametric expression (22) is satisfactory even at highextinction ratios such as r₁₀=13 dB.

Next, we explore the accuracy of parametric channel estimation (22) inIM/DD optical channels with combined Gaussian and ASE noise. We consideroptical channels with dispersion parameters D=1700, 3400, and 5100ps/nm, which correspond, for example, to 100 km (2^(δ)=8), 200 km(2^(δ)=32), and 300 km (2^(δ)=128) of SSMF, respectively. SSMF isStandard Single-Mode Fiber, as specified by the InternationalTelecommunications Union (ITU) Recommendation G.652. This is used in thethird telecommunications window (1550 nm), which leads to a dispersionparameter of 17 ps/nm/km. The data rate is 10 Gb/s and the transmittedpulse shape has an unchirped Gaussian envelope exp(−t²/2T₀ ²) with T₀=36ps. We assume that the MLSE-based receiver has enough states tocompensate the total memory of the channel (i.e., 2^(δ−1) states).

3.A. Measure of the Goodness of PDF Approximations for MetricsEvaluation

We introduce the average metric error (AME): a quantity that is bettersuited to assess the accuracy of branch-metric approximations in Viterbidecoders.

Let ƒ_(y|s)(y) and ƒ_(y|s)(y) be the true signal pdf and its estimate,respectively. We divide the interval of interest of y into N_(y)equidistant points Y={ y ^((k))}, k=1, . . . , N_(y). Then, we definethe AME as

$\begin{matrix}{{AME} = {\frac{1}{2^{\delta}N_{y}}{\sum\limits_{{\overset{\_}{s}}_{i} \in s}\;{\sum\limits_{{\overset{\_}{y}}^{(k)} \in y}\;{{{\ln\frac{{\hat{f}}_{y|s}\left( {\overset{\_}{y}}^{(k)} \middle| {\overset{\_}{s}}_{i} \right)}{f_{y|s}\left( {\overset{\_}{y}}^{(k)} \middle| {\overset{\_}{s}}_{i} \right)}}}.}}}}} & (30)\end{matrix}$Note that the AME is nonnegative and equal to zero if and only if{circumflex over (ƒ)}_(y|s)( y ^((k))| s _(i))=ƒ_(y|s)( y ^((k))| s_(i)), ∀ y ^((k))εY, ∀ s _(i)εS. Thus, the accuracy of the pdfapproximation improves as AME→0.

3.B. Channel Estimation in the Presence of Ideal Filters

Let S ₀ and S ₁ be the currents generated by each constellation symbolin a nondispersive optical channel ( S ₀+ S ₁=1). Then, we define

-   -   Extinction Ratio:

$\begin{matrix}{r_{0\; 1} = {r_{10}^{- 1} = \frac{{\overset{\_}{S}}_{0}}{{\overset{\_}{S}}_{1}}}} & (31)\end{matrix}$

-   -   Optical Signal-to-Noise Ratio:

$\begin{matrix}{{OSNR} = \frac{{\overset{\_}{S}}_{1}}{2\; I_{sp}}} & (32)\end{matrix}$

-   -   Signal-to-Gaussian-Noise Ratio:

$\begin{matrix}{{SGNR} = {\frac{1}{2^{\delta}\sigma_{e}^{2}}{\sum\limits_{{\overset{\_}{s}}_{i} \in s}^{\;}{\overset{\_}{s}}_{i}^{2}}}} & (33)\end{matrix}$

FIGS. 2A-2C shows the parameter v_(s) as a function of OSNR in anondispersive optical channel. We use y _(c)=5σ_(e), SGNR=20 dB, M=3,and several values of r₁₀. We present results derived from both theiterative method based on (23) (solid lines) and the approximation givenby (24) (“*”). In all cases, (24) is accurate in estimating parameterv_(s).

FIG. 3 shows the AME versus the OSNR for D=1700 ps/nm (2^(δ)=8), Y=[−2,5], N_(y)=2¹², M=3, two values of SGNR=20 dB and ∞, and three values ofr₁₀. The two sets of solid curves correspond to SGNR=20 dB, with theupper set corresponding to the Gaussian approximation and the lower setcorresponding to the parametric estimate (22). Each set of curvesincludes three curves, corresponding to three different values of r₁₀.In this and other figures, the curves for the parametric estimate (22)may be labeled as the new approach or the new channel estimate. The twosets of dotted curves correspond to SGNR=∞, again with the upper setcorresponding to the Gaussian approximation and the lower setcorresponding to the parametric estimate (22). In these calculations,the true pdf (as required to calculate the AME) is obtained by numericalintegration of (4). Parameters v_(s) and ζ_(s) are estimated asdescribed above.

FIGS. 4A and 4B show the received signal pdf's for the worst cases (wcs)found for the parametric estimate (22) and the Gaussian approximation,respectively. In all cases of FIGS. 3 and 4, the parametric estimate(22) out-performs the Gaussian approximation, particularly in thepresence of dominant ASE noise.

Similar results are observed in FIGS. 5A and 5B. FIG. 5A considers adispersive optical channel with D=3400 ps/nm (2^(δ)=32) and FIG. 5Bconsiders a dispersive optical channel with D=5100 ps/nm (2^(δ)=128). Inboth these figures, the solid curves correspond to SGNR=20 dB and thedotted curves to SGNR=∞. Again, the upper set of curves corresponds tothe Gaussian approximation and the lower set to the parametric estimate(22).

FIG. 6A shows AME versus the extinction-ratio r₁₀ for a nondispersivechannel with M=3, OSNR=12 dB, and SGNR=∞. We present results for boththe parametric channel-estimate (22) and the Gaussian approximation withthe true pdf given by (2). Two estimates for parameter v_(s) areconsidered: 1) v_(s)=v_(opt), where v_(opt) is the value that minimizesthe magnitude of the third-order central moment defined by (23) (“*”)and 2) approximation (24) (“^(o)”).

FIG. 6B shows BER versus r₁₀ for a perfect channel estimation and forthe parametric estimate (22). The good behavior of the latter withv_(opt) is verified from this figure. Note that the accuracy ofapproximation (24) degrades significantly at high values of r₁₀.Nevertheless, from FIG. 6B, we realize that approximation (24) can beused with satisfactory results in MLSE receivers over optical channelswith r₁₀≦10 dB.

3.C. Channel Estimation in the Presence of Ideal Filters andQuantization

Now consider the effects of quantization on the accuracy of the channelestimation. Let y′ and Δ be the quantized output and the quantizationstep (Δ∝2^(−L)), respectively. The pdf of y′ is discrete and equal tothe Δ-spaced samples of the smooth pdf of the signaly=x+z+q  (34)where x and z are defined in (1), and q is an independent uniformlydistributed random variable with pdf ƒ_(q)(q)=1/Δ, −Δ/2≦q≦Δ/2. Note thatx, z, and q are independent random variables, then the smooth pdf isƒ_(y|s)(y|s)=ƒ_(x|s)(y|s)

ƒ_(z)(y)

ƒ_(q)(y), sεS.  (35)

The discrete pdf of the quantized output y′ is given by the samples ofthe smooth pdf (35). Therefore, it is useful to investigate howaccurately the proposed method approximates (35).

FIGS. 7A-7B show the AME for different values of L with the true pdf(35). We consider D=5100 ps/nm (i.e., 300 km of SSMF), 2^(δ)=128,SGNR=20 dB and ∞ (FIGS. 7A and 7B, respectively), M=3, and r₁₀=10 dB.Parameters v_(s) are estimated as described previously with the centralmoments numerically evaluated from the smooth pdf (35). We note that theparametric estimate provides significant improvements in accuracy overthe Gaussian approximation for L=5 or higher.

This can also be observed in FIGS. 8A-8B, where we show pdf's for areduced set of the 2^(δ)=128 signal levels, with L=5, OSNR=12 dB, andSGNR=∞. From FIG. 7, we also observe that the accuracy of the parametricestimate degrades at high OSNR and low bit resolution. This is becausethe transformation T_(s)(•) cannot be assumed concave when the noise qis dominant (e.g., small value of L). Therefore, the accuracy of (21)degrades. Numerical results show that the generic pdf approximation (16)with the exact T_(s)(•) numerically computed from (10) and (35) achievesgood accuracy even for a number of bits as small as L=3. Notice that theGaussian approximation also yields the same poor accuracy in thepresence of dominant quantization noise.

However, we observe the following. An accuracy assessment of thereceiver performance is more important at low or medium OSNR than athigh OSNR. In this situation (e.g., OSNR≦16 dB and SGNR=∞), theparametric channel estimation with L=5 resolution achieves significantlybetter accuracy than the Gaussian approximation. The parametricestimate, unlike the Gaussian approximation, achieves good accuracy athigh OSNR if the resolution is sufficiently high (e.g., L≧6).

3.D. Channel Estimation in the Presence of Realistic Filters

The previous analysis assumes that the filters are ideal (e.g.,integrate-and-dump electrical filters) and that the ASE noise in theelectrical domain is a chi-square random variable. However, it is knownthat the accuracy of the chi-square model for the ASE noise pdf may bepoor in the presence of practical filters. Based on computersimulations, in the following, we explore the accuracy of the parametricchannel estimator in IM/DD systems with realistic optical/electricalfilters. The parametric channel estimator is satisfactory, even whenrealistic filters are considered. This can be inferred from the factthat in IM/DD optical systems with r₀₁>0 and practical filters, (5) isstill a good approximation for the pdf of the ASE noise in theelectrical domain. Therefore, the models and assumptions used above arealso valid.

In the following simulations, the parameters of the proposed expressionfor the pdf (16) are estimated from the sample moments of the inputsignal, as explained below. First, we consider a nondispersive opticalchannel: OOK NRZ modulation with r₁₀=13 dB, OSNR=15 dB, and SGNR=∞. Weuse a five-pole Bessel electrical filter with −3-dB bandwidth (3 dB-BW)of 7 GHz, and a Lorentzian optical filter with two −3-dB bandwidths: 10and 20 GHz. It has been shown that results for any arbitraryoptical-filter shape are generally between the Lorentzian and theideal-rectangular filter.

FIGS. 9A-9B depict the pdf's of the received signal obtained from (16),from computer simulations, and from the Gaussian approximation. Thenumber of samples used in the simulations is 5×10⁵. The excellentaccuracy of the parametric channel-estimation technique is verified. Onthe other hand, note that the Gaussian approximation fails in estimatingthe optimum-decision threshold, which would yield a significantperformance degradation.

FIG. 10 shows results for a dispersive optical channel with D=1700ps/nm, r₁₀=10 dB, four-state Viterbi decoder, SGNR=∞, OSNR=15 dB, andL=5. The signal samples at the output of the electrical filter are firstquantized and then used to estimate the parameters of the pdf (16), asexplained below. We use a five-pole Butterworth electrical filter with 3dB-BW=7 GHz and a Lorentzian optical filter with 3 dB-BW=20 GHz. Thenumber of samples used in the simulations is 5×10⁶. Again, from FIG. 10,the good accuracy of the parametric channel estimation with realisticoptical/electrical filters and finite-resolution analog-to-digitalconverters (ADCs) can be observed.

4. Performance of MLSE in IM/DD Optical Channels

Now consider the performance of MLSE in IM/DD optical channels based onthe new generic functional form for the pdf (16). Because of the OOKmodulation, a symbol error corresponds to exactly 1 bit error.Therefore, the probability of bit error of the Viterbi decoder is upperbounded by

$\begin{matrix}{P_{b} \leq {\sum\limits_{\Psi \neq \hat{\Psi}}^{\;}{{W_{H}\left( {\Psi,\hat{\Psi}} \right)}\Pr\left\{ {\hat{\Psi}❘\Psi} \right\}\Pr\left\{ \Psi \right\}}}} & (36)\end{matrix}$where Ψ={a_(n)} represents the transmitted sequence, {circumflex over(Ψ)}={â_(n)} is an erroneous sequence, Pr{{circumflex over (Ψ)}|Ψ} isthe probability of the error event that occurs when the Viterbi decoderchooses sequence {circumflex over (Ψ)} instead of Ψ), andW_(H)(Ψ,{circumflex over (Ψ)}) is the Hamming weight of Ψ XOR{circumflex over (Ψ)}, in other words, the number of bit errors in theerror event. Pr {Ψ} is the probability that the transmitter sentsequence Ψ.

It is possible to show that

$\begin{matrix}{{\Pr\left\{ {\hat{\Psi}❘\Psi} \right\}} \simeq {Q\left( {\sqrt{\sum\limits_{n = n_{0}}^{n_{1} + \delta - 1}\frac{1}{\varsigma_{s_{n}}}}❘{{{T_{s_{n}}\left( {\overset{\_}{y}}_{n} \right)} - {T_{s_{n}}\left( s_{n} \right)}}❘^{2}}} \right)}} & (37)\end{matrix}$where Q(x)=½erfc(x/√{square root over (2)}), and y=( y ₁, . . . , y_(N))εL(s, ŝ) is the vector such ū=T_(s)( y) minimizes F(u,s).

To compute an approximation to the receiver BER, (36) is used. As iscommon practice, the sum over error events in (36) is replaced by itslargest terms, whose values are approximated using (37).

4.A. Closed-Form Approximation for the Error-Event Probability inDispersive IM/DD Channels

We can derive a simple analytical approximation assumingT_(ŝ)(•)=T_(s)(•). From (37):

$\begin{matrix}{{\Pr\left\{ {\hat{\Psi}❘\Psi} \right\}} \simeq {{Q\left( {\sqrt{\sum\limits_{n = n_{0}}^{n_{1} + \delta - 1}{\frac{{T_{s_{n}}\left( {\hat{s}}_{n} \right)} - {T_{s_{n}}\left( s_{n} \right)}}{\sqrt{\varsigma_{s_{n}}} + \sqrt{{\overset{\sim}{\varsigma}}_{{\hat{s}}_{n}}}}}}}^{2} \right)}.}} & (38)\end{matrix}$

For Gaussian noise (N₀=0), from (25), we have v_(ŝ) _(n) =v_(s) _(n) =1and ζ_(s) _(n) ={tilde over (ζ)}_(s) _(n) =σ_(e) ²∀s_(n), ŝ_(n)εS.Therefore, (38) reduces to the well-known expression

$\begin{matrix}{{\Pr\left\{ {\hat{\Psi}❘\Psi} \right\}} \simeq {{Q\left( {\frac{1}{2\sigma_{e}}\sqrt{\sum\limits_{n = n_{0}}^{n_{1} + \delta - 1}{{{\hat{s}}_{n} - s_{n}}}^{2}}} \right)}.}} & (39)\end{matrix}$

For ASE noise (σ_(e)=0), from (25), we verify that v_(ŝ) _(n) =v_(s)_(n) =0.5 and ζ_(s) _(n) ={tilde over (ζ)}_(ŝ) _(n) =N₀/2∀s_(n),ŝ_(n)εS. Thus, from (44), we get

$\begin{matrix}{{\Pr\left\{ {\hat{\Psi}❘\Psi} \right\}} \simeq {{Q\left( \sqrt{\frac{1}{2\; N_{0}}{\sum\limits_{n = n_{0}}^{n_{1} + \delta - 1}{{\sqrt{{\hat{s}}_{n}} - \sqrt{s_{n}}}}^{2}}} \right)}.}} & (40)\end{matrix}$

For ASE noise (σ_(e)=0) and no dispersion (2^(δ)=2), from (40), weobtain

$\begin{matrix}\begin{matrix}{{\Pr\left\{ {\hat{\Psi}❘\Psi} \right\}} \simeq {Q\left( \sqrt{\frac{{{\hat{s} - s}}^{2}}{{{\sqrt{2\; N_{0}\hat{s}} + \sqrt{2\; N_{0}s}}}^{2}}} \right)}} \\{\approx {{Q\left( \sqrt{\frac{{{\hat{s} - s}}^{2}}{{{\sqrt{M_{2,\hat{s}}} + \sqrt{M_{2,s}}}}^{2}}} \right)}.}}\end{matrix} & (41)\end{matrix}$

Since Q(x)≈(1/√{square root over (2π)}x)e^(−(1/2)x) ² and assuming thatŝ and s are equiprobable, from (36) and (41), we get

$\begin{matrix}{P_{b} = {{\Pr\left\{ {\hat{\Psi}❘\Psi} \right\}} \approx {\frac{1}{\sqrt{2\;\pi}Q_{P}}{\mathbb{e}}^{{- \frac{1}{2}}Q_{P}^{2}}}}} & (42)\end{matrix}$where Qp=|ŝ−s|/(√{square root over (M_(2,s))}+√{square root over(M_(2,s))}) is the well-known Q factor. The good accuracy of (42) hasbeen verified when the intensity level for logical 0 does not vanish(e.g., r₀₁=0.1), and the OSNR is high (as we expressed above).

4.B. Numerical Results

We explore the accuracy of both the parametric channel estimation inMLSE-based receivers and the performance analysis developed in theprevious section. We present results for OOK RZ modulation. Data rate is10 Gb/s. The transmitted pulse shape has an unchirped Gaussian envelopeexp (−t²/2T₀ ²) with T₀=36 ps. We focus on IM/DD optical channels withcombined Gaussian and ASE noise. Channel estimation for these links hasbeen reported only for the case of ideal filters. Therefore, in order tocompare the parametric approach with previous work, ideal rectangularoptical and integrate-and-dump electrical filters are considered in thissection. It is important to realize, however, that results not includedhere for the case of realistic optical/electrical filters have shownthat the parametric approach achieves similar accuracy to that presentedhere.

FIGS. 11A-11C shows the BER versus OSNR in a nondispersive opticalchannel for M=3 and different values of SGNR and r₁₀. We present valuesderived from both the chi-square pdf's (solid lines) and the newapproach based on (38) (“*”). Function parameters are estimated asabove. In all cases, we verify the excellent accuracy of the valuesobtained from the performance analysis developed here.

Next, we analyze two optical channels with D=1700 and D=3400 ps/nm. TheMLSE receiver has enough states to compensate the total memory of thechannel (i.e., four- and 16-state Viterbi decoders for D=1700 and D=3400ps/nm, respectively). Exact knowledge of the channel impulse response isassumed. FIGS. 12A-12B shows BER versus OSNR for different values ofSGNR with M=3 and r₁₀=10 dB. We compare results from computer simulationfor the receiver that uses 1) the metric computation based on theiterative procedure proposed in “Calculation of bit-error probabilityfor a lightwave system with optical amplifiers and post-detectionGaussian noise,” J. Lightwave Technology, vol. 8 no. 12, previouslypresented. 505-513, April 1991; and 2) the metric defined by (17) withT_(s)(•) approximated by (21). Parameters are obtained from the channelestimator described above. Note the excellent agreement between valuesobtained from the simplified metric (“*”) and the iterative procedure(“o”). FIG. 12 also depicts theoretical bounds derived from (36) and(38) (solid lines). In this case, we use the terms in summation (36)that correspond to error events with W_(H) (Ψ, {circumflex over (Ψ)})≦4.In FIGS. 13A-13B, we show results for M=5 and M=10 with r₁₀=8 dB, D=1700ps/nm, and four-state Viterbi decoder. In all cases, comparisons betweenthe values derived from theory and simulations confirm the good accuracyof the analytical predictions based on (38).

5. Example Implementations

As described previously, the problem of channel estimation is importantin the implementation of MLSE-based EDC receivers. One advantage of theparametric channel estimator described above is that it results in asignificant complexity reduction for the receiver. Note that theparametric estimation technique can also be used to provide a prioriknowledge in combination with the histogram method.

The implementation that can make EDC most viable commercially iscurrently a digital monolithic integrated circuit in complementary metaloxide semiconductor (CMOS) technology. FIG. 14 shows a simplified blockdiagram of an EDC receiver for IM/DD optical channels at a data rate of10 Gb/s. Although the A/D converter may use an interleaved architectureand the DSP may use parallel processing, for simplicity, these detailsare omitted in FIG. 14. The optical signal is converted to a current bya p-i-n or avalanche photodetector 1410, and the resulting photocurrentis amplified and converted to a voltage by a transimpedance amplifier1420 and a linear postamplifier 1420. The resulting signal is low-passfiltered 1425 and level-adjusted by a programmable gain amplifier 1430and then converted to a digital representation by the A/D converter1440. Then, the signal is equalized by a feedforward equalizer 1450 andpassed to the branch-metric computation unit 1460. In some single-modefiber applications where the amount of dispersion is limited, the FFE1450 may be omitted.

Note that in the following example, referring to (1), the receivedsignal y_(n) is the input to the branch-metric computation unit (whichis also the FFE output in this case), and the various noise sources andthe “channel” account for all effects from the transmitter through theoptical fiber and including receiver effects prior to the branch-metriccomputation unit. The transmitted bit sequence a_(n) hopefully isrecovered by the decisions of the Viterbi decoder 1470.

Branch metrics can be efficiently computed using lookup tables (LUTs)1465. This assumes that the signal at the input of the Viterbi decoder1470 can be quantized to a relatively low resolution so that the size ofthe LUTs is reasonable. This assumption is valid in many practicalcases. For example, an MLSE-based receiver capable of compensating thechromatic dispersion and polarization mode dispersion of up to 300 km ofSSMF requires a resolution of about L=6 bits for this signal. In thiscase, the branch-metric LUTs 1465 have 64 entries per branch, a sizequite manageable in current CMOS technology. The contents of the LUTs1465 are channel-dependent and possibly also time-dependent. This is thecase, for example, of a single-mode fiber receiver that operates in thepresence of polarization mode dispersion. Therefore, the LUTs 1465preferably are implemented in random-access memory.

The function of the channel-estimator is to compute the contents of theLUTs 1465. To keep track of changes in the channel, the LUTs 1465 mustbe refreshed periodically. If the channel-estimation is based on aclosed-form parametric approach (such as (22)), the channel-estimatorwill be referred to as a parametric channel-estimator 1480.

In the example described above, the parametric channel-estimator 1480updates the LUTs 1265 according to the process shown in FIG. 15. In thisexample, the parametric channel-estimator 1480 is based on (22). In step1510, the parametric channel-estimator 1480 estimates the parameters forthe received signal pdf based on the received samples. In thisparticular example, the received samples are y_(n), the inputs to thebranch metric computation unit 1460. The parameters for (22) are v_(s),ζ_(s) and ū_(s) These parameters can be calculated for all signal levelss from the received samples y_(n) using the method of moments,specifically using (24), (26) and (27) above. In step 1520, theestimated parameters are used to evaluate the generic nonlineartransformation T_(s)(y) for each input value y_(n) and each signal levels. In step 1530, branch metrics are computed based on the transformedsignal T_(s)(y). In the implementation of FIG. 14, the parametricchannel estimator 1480 updates the LUTs 1465 based on the transformedsignal T_(s)(y). The parametric channel estimator 1480 preferably isimplemented by firmware or software executing on an embedded processor,although it could also be implemented by other types of hardware and/orsoftware.

Because channel-estimation algorithms are not very regular, they usuallydo not lend themselves well to be implemented in dedicated hardware.They are usually better implemented in firmware running on ageneral-purpose embedded processor. However, even a relatively fastprocessor can run out of time if a complicated channel-estimationalgorithm is used. This would be the case, for example, for algorithmsbased on iterative solutions of equations. It is important to realizethat the complexity of the channel-estimation algorithm grows whenadditional sources of noise, such as quantization noise, are taken intoaccount. Thus, it is clear that the computational load of 1) theestimation of the pdf signal parameters and 2) the refresh of thebranch-metric LUTs constitutes an important aspect of the design of theViterbi decoder that affects its implementation complexity and even itstechnical viability.

5.A. Practical Implementation Using the Method of Moments

To implement the parametric channel estimates of (22), estimates ofv_(s), ζ_(s), and s (or ū_(s)) are generally required for all 2^(δ)branches in the trellis. In the presence of quantization noise,parameters v_(s) can be expressed in terms of the moments of the smoothpdf (35). It has been shown that their values can be accuratelyapproximated by the moments of the quantized output y′ if the resolutionis sufficiently high (e.g., L>4 for the application considered above).Therefore, in practical implementation (i.e., realistic filters,finite-resolution A/D converters, etc.), the central moments required toevaluate v_(s) can be directly estimated from the sample moments of thereceived quantized samples y′.

Once the 2^(δ) values of v_(s), are estimated, from (10) through (16),we verify that the rest of the parameters required for metriccomputations reduce to means and variances of the random variable y_(n)^(v) ^(s) . These parameters can be 1) derived from (13) and (14) withT_(s)(y)=y^(v) ^(s) or 2) estimated from the sample moments:

$\begin{matrix}\begin{matrix}{{\overset{\_}{u}}_{s} = {E_{s}\left\{ y_{n}^{v_{s}} \right\}}} \\{\approx {\frac{1}{N_{m}}{\sum\limits_{k = n}^{n + N_{m} - 1}y_{k}^{v_{s}}}}}\end{matrix} & (43) \\\begin{matrix}{\varsigma_{s} = {E_{s}\left\{ \left( {y_{n}^{v_{s}} - {\overset{\_}{u}}_{s}} \right)^{2} \right\}}} \\{{\approx {\left( {\frac{1}{N_{m}}{\sum\limits_{k = n}^{n + N_{m} - 1}\left( y_{k}^{v_{s}} \right)^{2}}} \right) - {\overset{\_}{u}}_{s}^{2}}},{s \in {S.}}}\end{matrix} & (44)\end{matrix}$

In practical implementations on integrated circuits, the function y^(v)^(s) is efficiently computed using LUTs, which can be generated from theN_(v)-order Taylor's expansion of y^(v) ^(s) given by

$\begin{matrix}{y^{v_{s}} \approx {1 + {\sum\limits_{k = 1}^{N_{v}}{\frac{1}{k!}{\left( {v_{s}\;{\ln(y)}} \right)^{k}.}}}}} & (45)\end{matrix}$

Further simplification can be achieved by using well-knownfunction-evaluation techniques. For example, if we writev_(s)=v_(ref)+v_(d,s), then y^(v) ^(s) =y^(v) ^(ref) y^(v) ^(d,s) . Ifwe tabulate y^(v) ^(ref) , then the size of v_(d,s) can be reduced.Thus, only a few terms of the series are needed to evaluate y^(v) ^(d,s). In particular, we have found that

$\begin{matrix}{y^{v_{s}} \approx {y^{v_{ref}}\left\lbrack {1 + {v_{d,s}\;{\ln(y)}} + {\frac{1}{2}\left( {v_{d,s}\;{\ln(y)}} \right)^{2}}} \right\rbrack}} & (46)\end{matrix}$with v_(ref)=0.7 providing excellent accuracy for the v_(s) range ofinterest in the application considered above (i.e., v_(s)ε(0.3,1]). Thesets of 2^(L) values for ln(y) and y^(v) ^(ref) can be stored inread-only memory (ROM), and thus, the 2^(δ) LUTs (one for each v_(s))can be easily generated from (46). Note that this operation can proceedat a low speed. Therefore, it can be carried out by an embeddedgeneral-purpose processor. Once the 2^(δ) LUTs with 2^(L) words fory^(v) ^(s) are generated, the 2^(δ) values of ū_(s) and ζ_(s) can beeasily calculated.

Estimators of IM/DD optical channels proposed in the past are based onthe method of steepest descent. Comparisons of computational complexityindicate that the parametric approach described above requiresapproximately 5% of the computational load required by the steepestdescent approach. Consider an eight-state Viterbi decoder with L=6resolution bits. Taking into account that typical performance ofembedded general-purpose processors allowed by current technology (e.g.,90-nm CMOS process) is in the 500-900 MIPS range, it is concluded thatthe implementation of channel estimators based on the steepest descentmethod is seriously limited. On the other hand, we verify that theparametric channel estimator proposed above could be easily implementedby using current technology. The parametric approach is very attractivefor practical implementations of high-speed MLSE-based IM/DD receiversin integrated circuits in CMOS technology.

6. Further Examples

The examples described above were based on IM/DD optical channels. Thisexample was chosen partly because there is currently significantinterest in this application. It was chosen also partly because certainprinciples are more easily described using a specific example. However,the invention is not limited to this particular example. The principlescan be extended to communications channels other than IM/DD opticalchannels.

Other examples include read channels for magnetic recording. Otherchannels where there may be a mix of Gaussian and non-Gaussian noise arechannels subject to crosstalk. Crosstalk, in general, does not have aGaussian distribution. Some examples of channels that suffer fromcrosstalk, in addition to other forms of noise, are twisted pairchannels, such as those specified by the IEEE standards 1000BASE-T and10GBASE-T, the channel specified by the SFP+ standard, DWDM opticalchannels where nonlinear intermodulation effects among differentwavelengths (including four-wave mixing and cross-phase modulation)result in optical crosstalk coupling to the channel of interest, etc.Another important example of non-Gaussian noise is cochannelinterference in wireless channels. In these cases, the receiver alsooperates in the presence of a mixture of Gaussian and non-Gaussiannoise. In these examples, the specific closed-form parametric expression(22) may not be applicable, but the general approach described abovebased on nonlinear transformation T_(s)(y) may result in other usefulclosed-form parametric expressions.

As another example, receivers using the approach described above are notrequired to use a Viterbi decoder or to expressly calculate branchmetrics. The decision-feedback equalizer (DFE), the maximum a posteriori(MAP) decoder, and iterative decoding receivers are examples ofnon-Viterbi decoders that can benefit from a closed-form parametricmodel of the channel.

Finally, the various components shown in block diagrams are not meant tobe limited to a specific physical form. Depending on the specificapplication, they can be implemented as hardware, firmware, software,and/or combinations of these. In addition, the “coupling” betweencomponents may also take different forms. Dedicated circuitry can becoupled to each other by hardwiring or by accessing a common register ormemory location, for example. Software “coupling” can occur by anynumber of ways to pass information between software components (orbetween software and hardware, if that is the case). The term “coupling”is meant to include all of these and is not meant to be limited to ahardwired permanent connection between two components. In addition,there may be intervening elements. For example, when two elements aredescribed as being coupled to each other, this does not imply that theelements are directly coupled to each other nor does it preclude the useof other elements between the two.

Although the detailed description contains many specifics, these shouldnot be construed as limiting the scope of the invention but merely asillustrating different examples and aspects of the invention. It shouldbe appreciated that the scope of the invention includes otherembodiments not discussed in detail above. Various other modifications,changes and variations which will be apparent to those skilled in theart may be made in the arrangement, operation and details of the methodand apparatus of the present invention disclosed herein withoutdeparting from the spirit and scope of the invention as defined in theappended claims. Therefore, the scope of the invention should bedetermined by the appended claims and their legal equivalents.

What is claimed is:
 1. A receiver for determining a received bitsequence based on a received signal transmitted to the receiver over achannel, the receiver comprising: a parametric channel-estimator thatprovides a channel-estimate based on a closed-form parametric model ofthe channel, the parametric channel-estimator estimating parameters forthe parametric model based on the received signal, wherein theclosed-form parametric model of the channel is based on a nonlineartransformation T_(s)(y) as defined in the following equation:u=T _(s)(y)=F _(u|s) ⁽⁻¹⁾(F _(y|s)(y)), where s is a noise-free signal,u is a Gaussian random variable, T_(s) is a transformation function, yis a random variable representing a received input symbol, F_(u|s)(•) isa cumulative distribution function of u when the noise-free signal s isreceived, F_(y|s)(•) is a cumulative distribution function of y when thenoise-free signal s is received; and a decoder coupled to the parametricchannel-estimator, the decoder determining the received bit sequencebased in part on the channel-estimate from the parametricchannel-estimator.
 2. The receiver of claim 1 where there is noclosed-form exact model of the channel.
 3. The receiver of claim 1wherein the closed-form parametric model of the channel is based on thefollowing equation:${{f_{y❘s}\left( {y❘s} \right)} = {\frac{1}{\sqrt{2\;\pi\;\varsigma_{s}}}{\mathbb{e}}^{- {\frac{1}{2\;\varsigma_{s}}{\lbrack{{T_{s}{(y)}} - {\overset{\_}{u}}_{s}}\rbrack}}^{2}}{T_{s}^{\prime}(y)}}},{\forall{y.}}$wherein ƒ_(y|s)(y|s)is a probability density function for a receivedsignal y given a noise free signal s, u is a Gaussian random variablesuch that u=T_(s)(y)=F_(u|s) ⁽⁻¹⁾(F_(y|s)(y)) where T_(s)(y) is atransformation function for transforming y to u, F_(u|s)(•) is acumulative distribution function of u when a noise-free signal s isreceived, F_(y|s)(•) is a cumulative distribution function of y when thenoise-free signal s is received, T′_(s)(y)=dT_(s)(y)/dy, ū_(s) is themean of u, and ζ_(s) is a variance of u.
 4. The receiver of claim 1wherein the closed-form parametric model of the channel is based on thefollowing equation:${{f_{y❘s}\left( {y❘s} \right)} \approx {\frac{1}{\sqrt{2\;\pi\; M_{2,s}}}{\mathbb{e}}^{- {\frac{1}{2\;\varsigma_{s}}{\lbrack{{T_{s}{(y)}} - {T_{s}{(s)}}}\rbrack}}^{2}}s}} \in S$wherein ƒ_(y|s)(y|s) is a probability density function for a receivedsignal y given a noise free signal s, T_(s)(•) is a transformationfunction for transforming y to a Gaussian random variable u asu=T_(s)(y)=F_(u|s) ⁽⁻¹⁾(F_(y|s)(y)) where F_(u|s)(•) is a cumulativedistribution function of u when a noise-free signal s is received, andF_(y|s)(•) is a cumulative distribution function of y when thenoise-free signal s is received, ζ_(s) is a variance of u and M_(2,s) isa conditional second-order central moment of y.
 5. The receiver of claim1 wherein the parameters include {v_(s)} and {ζ_(s)} with sεS, whereinζ_(s) is a variance of a Gaussian random variable and wherein v_(s) isin a range 0<v_(s)≦1.
 6. The receiver of claim 5 wherein the parametricmodel is defined by${{f_{y❘s}\left( {y❘s} \right)} = {\frac{v_{s}s^{({v_{s} - 1})}}{\sqrt{2\;\pi\;\varsigma_{s}}}{\mathbb{e}}^{{- \frac{1}{2\varsigma_{s}}}{({y^{v_{s}} - s^{v_{s}}})}^{2}}}},$sεS wherein f_(y|s)(y|s) is a probability density function for areceived signal y given a noise free signal s, ζ_(s) is a variance of aGaussian random variable, and wherein v_(s) is in a range 0<v_(s)≦1. 7.The receiver of claim 5 wherein the parametric channel-estimatorestimates {v_(s)} and {ζ_(s)}, wherein {v_(s)} is a value that minimizesa third order central moment of a Gaussian random variable, and whereinζ_(s) approximates a variance of the Gaussian random variable.
 8. Thereceiver of claim 1 wherein the channel includes non-Gaussian noise. 9.The receiver of claim 1 wherein the channel includes ASE noise.
 10. Thereceiver of claim 1 wherein the channel includes quantization noise. 11.The receiver of claim 1 wherein the channel includes signal-dependentnoise.
 12. The receiver of claim 1 wherein the channel is subject tocrosstalk.
 13. The receiver of claim 1 wherein the channel includesnonlinear intermodulation effects among different wavelengths within thechannel.
 14. The receiver of claim 1 wherein the channel includescochannel interference.
 15. The receiver of claim 1 wherein theparametric model is sufficiently accurate to account for at least twodifferent types of noise.
 16. The receiver of claim 15 wherein theparametric model is sufficiently accurate to account for Gaussian noiseand to account for ASE noise.
 17. The receiver of claim 1 wherein thechannel includes an optical fiber and the bit sequence is transmitted ata speed of at least 1 Gb/s.
 18. The receiver of claim 1 wherein thechannel includes an optical fiber and the bit sequence is transmitted ata speed of at least 10 Gb/s.
 19. The receiver of claim 18 wherein thechannel suffers from chromatic dispersion and polarization-modedispersion.
 20. The receiver of claim 19 wherein the channel hasdispersion of at least 1200 ps/nm.
 21. The receiver of claim 18 whereinthe received signal is transmitted using anintensity-modulation/direct-detection system.
 22. The receiver of claim21 wherein the received signal uses on-off keying modulation.
 23. Thereceiver of claim 1 wherein the receiver is an MLSE receiver.
 24. Thereceiver of claim 1 wherein the decoder is a Viterbi decoder.
 25. Thereceiver of claim 1 wherein the decoder includes a decision feedbackequalizer (DFE).
 26. The receiver of claim 1 wherein the decoder is amaximum a posteriori (MAP) decoder.
 27. The receiver of claim 1 whereinthe receiver is an iterative decoding receiver.
 28. The receiver ofclaim 1 further comprising: a branch-metric computation unit coupled tothe parametric channel-estimator, the branch-metric computation unitdetermining branch metrics for each of the possible received bitsequences based in part on the channel-estimate from the parametricchannel-estimator; wherein the decoder determines the received bitsequence based in part on the branch metrics from the branch-metriccomputation unit.
 29. The receiver of claim 28 wherein the branch-metriccomputation unit comprises a plurality of lookup tables for determiningthe branch metrics and the parametric channel-estimator updates thelookup tables based on the parametric model and the estimatedparameters.
 30. The receiver of claim 1 further comprising: an equalizercoupled to the branch-metric computation unit, that provides thereceived signal to the branch-metric computation unit.
 31. The receiverof claim 1 wherein the parametric channel-estimator, branch-metriccomputation unit and decoder are implemented as a single CMOS integratedcircuit.
 32. The receiver of claim 1 wherein the parametricchannel-estimator, branch-metric computation unit and decoder areimplemented as a single DSP integrated circuit.
 33. The receiver ofclaim 1, wherein the receiver comprises a direct detection receiver. 34.The receiver of claim 1, wherein T_(s)(y) is approximated by y^(v) ^(s)where 0<v_(s)≦1.
 35. The receiver of claim 34, wherein v_(s)=0.5.
 36. Amethod for determining a received bit sequence based on a receivedsignal transmitted over a channel, the method comprising: modeling thechannel using a closed-form parametric model defined by parameters,wherein the closed-form parametric model of the channel is based on anonlinear transformation T_(s)(y) as defined in the following equation:u=T_(s)(y)=F_(u|s) ⁽⁻¹⁾(F_(y|s)(y)), where s is a noise-free signal, uis a Gaussian random variable, T_(s) is a transformation function, y isa random variable representing a received input symbol, F_(u|s)(•) is acumulative distribution function of u when the noise-free signal s isreceived, F_(y|s)(•) is a cumulative distribution function of y when thenoise-free signal s is received; estimating the parameters for theparametric model based on the received signal, thereby providing achannel-estimate; determining the received bit sequence based in part onthe channel-estimate.
 37. The method of claim 36, wherein theclosed-form parametric model of the channel is based on the followingequation:${{f_{y❘s}\left( {y❘s} \right)} = {\frac{1}{\sqrt{2\;\pi\;\varsigma_{s}}}{\mathbb{e}}^{- {\frac{1}{2\;\varsigma_{s}}{\lbrack{{T_{s}{(y)}} - {\overset{\_}{u}}_{s}}\rbrack}}^{2}}{T_{s}^{\prime}(y)}}},{\forall{y.}}$wherein ƒ_(y|s)(y|s) is a probability density function for a receivedsignal y given a noise free signal s, u is a Gaussian random variablesuch that u=T_(s)(y)=F_(u|s) ⁽⁻¹⁾(F_(y|s)(y)) where T_(s)(y) is atransformation function for transforming y to u, F_(u|s)(•) is acumulative distribution function of u when a noise-free signal s isreceived, F_(y|s)(•) is a cumulative distribution function of y when thenoise-free signal s is received, T′_(s)(y)=dT_(s)(y)/dy, ū_(s) is themean of u, and ζ_(s) is a variance of u.
 38. The method of claim 36wherein the closed-form parametric model of the channel is based on thefollowing equation:${{f_{y❘s}\left( {y❘s} \right)} \approx {\frac{1}{\sqrt{2\;\pi\; M_{2,s}}}{\mathbb{e}}^{- {\frac{1}{2\;\varsigma_{s}}{\lbrack{{T_{s}{(y)}} - {T_{s}{(s)}}}\rbrack}}^{2}}s}} \in S$wherein ƒ_(y|s)(y|s) is a probability density function for a receivedsignal y given a noise free signal s, T_(s)(•) is a transformationfunction for transforming y to a Gaussian random variable u asu=T_(s)(y)=F_(u|s) ⁽⁻¹⁾(F_(y|s)(y)) where F_(u|s)(•) is a cumulativedistribution function of u when a noise-free signal s is received, andF_(y|s)(•) is a cumulative distribution function of y when thenoise-free signal s is received, ζ_(S) is a variance of u and M_(2,s) isa conditional second-order central moment of y.
 39. The method of claim36 wherein the parameters include {v_(s)} and {ζ_(s)} with sεS, whereinζ_(S) is a variance of a Gaussian random variable and wherein v_(s) isin a range 0<v_(s)≦1.
 40. The method of claim 39 wherein the parametricmodel is defined by${{f_{y❘s}\left( {y❘s} \right)} = {\frac{v_{s}s^{({v_{s} - 1})}}{\sqrt{2\;\pi\;\varsigma_{s}}}{\mathbb{e}}^{{- \frac{1}{2\varsigma_{s}}}{({y^{v_{s}} - s^{v_{s}}})}^{2}}}},$sεS wherein ƒ_(y|s)(y|s) is a probability density function for areceived signal y given a noise free signal s, ζ_(s) is a variance of aGaussian random variable, and wherein v_(s) is in a range 0<v_(s)≦1. 41.The method of claim 36, further comprising receiving the received signalusing a direct detection receiver.
 42. The method of claim 36, whereinT_(s)(y) is approximated by y^(v) ^(s) where 0<v_(s)≦1.
 43. The methodof claim 42, wherein v_(s)=0.5.